Rain delayed the game nearly three hours in the bottom of the fifth inning, which ended the starters' outings earlier than planned.
Boston's Alfredo Aceves went five innings, allowing only three hits and one run.
Chris Archer threw 4.2 innings for the Rays, surrendering five hits, walking five batters and giving up four runs in the process.
The Red Sox led 4-1 at the time of the delay and added an insurance run in the sixth inning when Shane Victorino tripled to to right field.
Tampa Bay has now dropped five of its last six games.
The first game of the doubleheader marked the MLB debut of Wil Myers, one of Tampa Bay's top prospects. Myers was hitless in four at-bats with one strikeout at Fenway Park.
The nightcap of this AL East doubleheader is important for both teams since the top four teams in the division are so close to each other in the standings.
Doubront pitched five innings and ultimately earned a no-decision as the Red Sox won in Tampa Bay on May 16. He is 2-1 in his career against Tampa Bay with a 3.91 ERA.
Boston has lost five of the last eight games in which the Venezuelan lefty has started.
Odorizzi is in his first season with the Rays, and he will make his fifth career start tonight. The 23-year-old native of Breese, Ill. has allowed 17 hits and 11 earned runs in 12.1 innings this season. Tonight will be his first appearance against the Boston Red Sox and only his third time pitching on the road.
The first pitch for Game 2 is scheduled for 8:05 p.m. ET.
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